Survival Analysis of Tensile Strength Variation and Simulated Length–Size Effect along Oak Boards

Tapia, Cristóbal; Aicher, Simon

doi:10.1061/(ASCE)EM.1943-7889.0002006

Open access

Technical Papers

Nov 2, 2021

Survival Analysis of Tensile Strength Variation and Simulated Length–Size Effect along Oak Boards

Authors: Cristóbal Tapia https://orcid.org/0000-0003-2228-1686 [email protected] and Simon Aicher https://orcid.org/0000-0003-1759-9884Author Affiliations

Publication: Journal of Engineering Mechanics

Volume 148, Issue 1

https://doi.org/10.1061/(ASCE)EM.1943-7889.0002006

PDF

Abstract

Experiments and a simulation of the effect of localized tensile strength in European white oak boards and predictions of the effect of the board’s length on global tensile strength are reported. Furthermore, the cross-correlation of local strength with a local modulus of elasticity (MOE) is investigated. This knowledge is a fundamental prerequisite for stochastic simulations of glued laminated timber with adequate representation of the lamination effect. The sample consisted of 47 boards, each subdivided virtually into 15 consecutive cells 100 mm in length and for which local MOE and its autocorrelation had been previously determined. The remnants of the first, that is, global, failure in the axial tension tests were loaded to rupture in secondary loadings up to four times. This procedure, resulting in a total of 102 primary and secondary strengths, was enabled by the rather blunt failure planes presented by the investigated oak boards. The censored localized strength data, including the lower bounds of the unbroken virtual cells, were analyzed using survival analysis with a modified likelihood function (LHF) applied to six different strength distribution models: (1) four censored parametric distributions (two-parameter Weibull, beta, grafted Weibull–Gaussian, and Duxbury–Leath–Beale); and (2) two censored regression models (Weibull and beta) with shape and scale parameters depending on global MOE. A detailed analysis of the lower tail of the experimental distribution evidenced the quasibrittle nature of the material, thus supporting the use of the Weibull–Gaussian distribution. The simulation of the tensile strength variation along the board’s length was conducted by an adapted vector autoregressive (VAR) model. Here, the localized MOEs were cross-correlated with the obtained cell-related strength distributions employing Gaussian normalization of both distributions, followed by mapping the normalized strength values into the LHF-derived strength distributions. The length effect of tensile strength was analyzed with virtual boards of different lengths with 50,000 simulations per board length and a strength distribution model. The obtained length effect exponent of a simple power law, usually applied in timber engineering, was roughly equal to 0.33 for the mean strength level for all models and grade combinations. At the 5%-quantile level, the calculated exponents varied and were grade-dependent and model-dependent from 0.14 to 0.34. The simulated length effects were similar to those of softwoods at the 5%-quantile level but markedly higher at the mean level, which is well explained by the finite weakest link theory by considering the more general size dependency that governs quasibrittle materials.

Introduction

The study of the variation of mechanical properties along boards has been a recurrent subject since the beginning of the 1980s. The fluctuation in the modulus of elasticity (MOE), as well as in the tensile and compressive strengths along boards, has mostly been investigated in the context of numerical strength models for glued laminated timber (GLT) beams, for which a descriptive model for these variables is needed as input. Foschi and Barrett (1980) initiated this field of material science with a simulation model for the variation of MOE and tensile strength of Douglas fir boards based on Weibull’s weakest link (WL) theory (Weibull 1951). Investigations on the variability of mechanical properties of spruce in cells 150 mm in length by Ehlbeck et al. (1984) set the foundations for what is known as the Karlsruher Rechenmodel—a numerical finite-element–based GLT strength model. This model was reimplemented by Frese (2006), when it was calibrated for GLT beams of the hardwood species beech.

Investigations by Kline et al. (1986) and Taylor and Bender (1988) added the spatial component to the variation analysis by considering the lag-correlations of both MOE and tensile strength. For this, boards were subdivided into segments of constant lengths, where the studied variables were measured. The obtained results were then used to calibrate an autoregressive (AR) model for the variables, later implemented in a GLT strength model (Taylor and Bender 1991). A related model was presented by Lam and Varoglu (1991b) for tensile strength profiles of spruce–pine–fir specimens for segments 610 mm in length. Based on experimental results (Lam and Varoglu 1991a), a moving average process of order 3 [MA(3)] was then developed.

Isaksson (1999) focused on the variation of edgewise bending strength in spruce boards and developed a model based on the concept of weak zones, defined as those areas in a board that contained knots of significant size (defined by a relative threshold). In essence, the model considered the alternate arrangement of weak and strong sections along boards. Strong sections were attributed throughout to the same strength value in each individual board. The strength of weak sections and the distance between each consecutive weak section followed a stochastic model. This approach was then adopted by Fink (2014) in his study of spruce boards, in which a censored regression analysis was also used to estimate the parameters governing the variation of strength along the board.

A rather neglected aspect of the variation of tensile strength in timber boards in general relates to the theoretical framework considered when analyzing the experimental strength distributions. Although it is common to assume WL theory for the study of timber members, thus a Weibull distribution, there is enough scientific evidence pointing at the inapplicability of this theory to quasibrittle materials, such as wood [see, e.g., Pang et al. (2008), Bažant and Le (2017)]. Although the weakest link concept, that is, structural failure triggered by the failure of the weakest element, applies generically, the Weibull distribution requires a very large, in fact infinite, number of links. This condition is met by perfectly brittle materials because their fracture process zone (FPZ) is negligible compared to the size of the structure. Thus, the number of representative volume elements (RVEs) is large enough for the strength to be approximated by the Weibull distribution. However, for quasibrittle materials, having a FPZ of nonnegligible size, the relatively low number of RVEs prevents the full development of the Weibull distribution, which is constrained to the lower tail of the distribution. On the opposite side of the failure spectrum, ductile failures occur, which are governed by the simultaneous mobilization of the strengths of all RVEs on the failure surfaces. This implies that the failure load is a weighted sum of each of these strengths, which, by the central limit theorem of probability, must converge to the Gaussian distribution (Pang et al. 2008). It has been experimentally shown (Bažant and Le 2017) that quasibrittle materials move from a predominantly Gaussian behavior, for relatively small sizes, to a predominantly Weibull behavior as the relative size increases, thus bridging plastic and brittle failure.

Bažant and Pang (2007) considered this duality and developed the so-called finite weakest-link (FWL) theory—emphasizing the difference with the typical infinite weakest link theory—which introduces a smooth transition between plastic and brittle failure and is best described by a grafted Weibull–Gaussian distribution (discussed subsequently). The shortcomings of the Weibull theory for quasibrittle materials were previously addressed by Duxbury et al. (1987), too, who derived a statistical distribution for the strength of quasibrittle materials based on a fuse network model. This model has been proven to accurately describe experimental data (van den Born et al. 1991) and simulation results (Bertalan et al. 2014).

In the first step of an experimental campaign on European oak boards (Quercus petraea, Q. robur), the MOE variation along the boards was investigated (Tapia and Aicher 2018, 2019). The free length of oak boards, tested in tension, was virtually subdivided into 15 segments (cells) 100 mm in length each, and the local MOE of each cell was measured. The MOE variation was studied based on serial correlation analysis and is explained in detail in Tapia and Aicher (2021b). In a second step, the boards were tested in tension until failure, and the remnants were tested, if possible, in secondary tests until failure.

This paper presents an analysis of the multiple fracture tests, which is a prerequisite for a stochastic GLT strength model. The data of the measured local tensile strengths were analyzed using survival analysis, which permits accounting for the missing information in the data set, that is, for the tensile strength of the unbroken cells. The estimated parameters for the different models were then used to simulate tensile strength profiles along oak boards, which in turn allows accounting for the correlation between MOE and tensile strength profiles during the simulation process. Finally, the stochastic cell model was employed to simulate the effect of the board’s length on its global tensile strength.

Materials and Methods

Oak Boards, Grading, Global and Local MOE

Investigations into MOE variations, tensile strength, and density were performed using European oak boards (Q. robur or Q. petraea), originating from the southwestern part of France. The sample consisted of 52 boards and contained a mixture of appearance grades QF2 and QF3, according to EN 975-1 (CEN 2009). The two QF appearance grades intentionally enabled a wide range of knottiness and grain deviation in the sample and resembled the material used by Faye et al. (2017) on an investigation of GLT beams of oak. The nominal dimensions (

length l \times width w \times thickness t

) of the planed boards were

2,500 \times 175 \times 24 mm

, respectively. The moisture content (MC) was, on average, 10.2% (coefficient of variation

[COV] = 4.6 %

). The density adjusted to an MC of 12% was

699 \pm 47 kg / m^{3}

. Thereafter, the sample was visually graded at the Materials Testing Institute (MPA), University of Stuttgart, into hardwood strength grades LS7, LS10, and LS13, as specified in the German structural hardwood grading standard DIN 4074-5 (DIN 2008). The details of the strength grading, together with exact quantification and allocation of the growth defects of each board, are given in Tapia and Aicher (2019).

The global modulus of elasticity was measured by axial tensile loading on a length

l_{E} = 8.6 \cdot w = 1,500 mm

, resulting in a mean and standard deviation for the entire set of

11.6 \pm 2.1 GPa

when corrected to 12% MC. For the subset of LS10 and LS13 boards, the MOE was

11.8 \pm 2.0 GPa

. The length employed to measure the MOE

l_{E}

differs from

l_{e} = 5 \cdot w

stipulated in EN 408 (CEN 2012), owing to the specific objective of the tests: measuring the variation of the MOE along the longest possible free length. For determining the local tensile MOE variation along the boards’ length

l_{E}

, each board was virtually subdivided into cells of 100 mm in length (Fig. 1), for which the MOE was measured in a discrete manner (Tapia and Aicher 2021b). The cell length of 100 mm was chosen for two reasons: (1) it is very close to the evaluation length of 90 mm, for which Olsson and Oscarsson (2017) found the best correlation between localized MOE based on fiber orientation measurements and bending strength, and somewhat shorter than the cell size used, among others, by Ehlbeck et al. (1984) and Frese (2006) for their strength models, and (2) it was the minimum length achievable by modifying an existing extensometer used to measure the localized MOE in the same boards (Tapia and Aicher 2021b). Because the objective was to study the localized MOE and tensile strength variation jointly, the same cell size of 100 mm was used to investigate the localized tensile strength, too.

Fig. 1. Measurement of multiple tensile strength values per board: (a) first test: free length $l_{s}$ ; and (b) secondary tests: free lengths $l_{s, 2, 1}$ , and $l_{s, 2, 2}$ .

Multiple Strength Measurements

The boards were tested in tension parallel to the grain direction until failure at a constant piston displacement rate of

0.04 mm / \min

with a computer-controlled servo-hydraulic testing machine. Both ends of each board were fixed by hydraulically actuated grips, which gradually introduced the clamping pressure along the clamping length (

\approx 350 mm

), minimizing stress concentrations at the transition of free length to the clamps. Each board’s first testing to failure delivered the global tensile strength

f_{t, 0, glob}

corresponding to a free length

l_{s}

of

1620 mm = 9.25 \cdot w

, which conforms closely to the provisions of EN 408 (CEN 2012) and EN 384 (CEN 2018), where

l_{s} = 9 \cdot w

is stipulated. Because the employed free length was slightly (3%) longer than specified in the European standards, the obtained global tensile strength values can be regarded as being conservative to a very small degree, which means that the obtained strength values tended to be slightly lower (about 1%) as compared to the standard approach. The mentioned strength decrease resulted from the fact that there is a higher probability of more and larger growth-bound defects (within the limits of the respective grading or strength class) in an increased free test-length, as compared to shorter ones. The stated quantification of the tensile strength reduction resulting from the increased free test length is based on the following length effect simulation results.

The remaining parts of each broken board left after the first global ultimate failure tests were tested, if possible, in a second and occasionally third and fourth tensile test, similar to those done by Lam and Varoglu (1991a), then for spruce–pine–fir boards of 6.1 m in length. The location, that is, the number of the cell responsible for the failure, was recorded. The procedure of sampling multiple tensile strength values from one board is illustrated in Fig. 1. The outlined procedure was fostered by the predominant occurrence of blunt failures of the tested oak boards, producing two remaining parts separated by the first (global) failure by fracture planes rather perpendicular to the board’s length axis [Figs. 2(a and b)]. The free lengths (

l_{s, 2, i}

) between the grips of these secondary tensile tests were significantly lower than the primary global test. The values for

l_{s, 2, i}

were on average (

\pm std.

)

(3.2 \pm 1.7) \cdot w

with a minimum of

0.6 \cdot w

. However, the reduced lengths of the remnants do not constitute a methodological problem because the secondary tests are intended to detect the strengths of the stronger parts of the boards in relation to the weakest section in the board to obtain a larger database for local MOE, density, and strength correlations.

Fig. 2. Examples of blunt failures obtained in the tensile tests parallel to the fiber of the oak boards.

Finite Weakest Link Theory

The finite weakest-link theory deals with the statistical characterization of the failure behavior of quasibrittle materials and was developed in a series of works by Bažant and Pang (2006, 2007), and Le et al. (2011), among others. The theory describes the transition zone between ductile and brittle behavior by considering a power law in the lower tail of the strength distribution (brittle failure) combined with a normally distributed core (ductile failure). Such behavior is accurately represented by the so-called grafted Weibull–Gaussian (WG) distribution (Le and Bažant 2011; Bažant and Le 2017) as given in Table 1. There,

x_{gr}

is the grafting point,

m

and

δ

are the shape and scale parameters of the two-parameter Weibull distribution, and

μ

and

σ

are the mean and standard deviation of the Gaussian core. The grafting probability

F (x_{gr}) = F_{gr}

is defined as

F_{gr} = 1 - \exp [- {(\frac{x_{gr}}{δ})}^{m}]

(1)

Table 1. Statistical distributions and fitted parameters for the different investigated models to describe the tensile strength variation along board length

Model		Parameters^a
Censored parametric models:
(a)	Weibull	$(δ, m)$
(b)	Beta	$(λ, δ, a, b)$
(c)	Weibull–Gaussian	$(μ, σ, δ, m)$
(d)	DLB^b	$(δ, κ, ρ)$
Censored regressions:
(e)	Weibull	$(α_{0}, α_{1}, β_{0}, β_{1})$
(f)	Beta	$(α_{0}, α_{1}, β_{0}, β_{1}, γ_{0}, γ_{1})$
	with:	$λ = 0$
		$δ = \exp (α_{0} + α_{1} \cdot E_{t, 0, glob})$
		$m, a = \exp (β_{0} + β_{1} \cdot E_{t, 0, glob})$
		$b = \exp (γ_{0} + γ_{1} \cdot E_{t, 0, glob})$
Distributions:
	Weibull:	$F (x) = 1 - \exp [- {(\frac{x}{δ})}^{m}]$
	Beta:^c^,^d	$f (x) = \frac{Γ (a + b)}{Γ (a) Γ (b)} \cdot {(\frac{x - λ}{δ})}^{a - 1} {(1 - \frac{x - λ}{δ})}^{b - 1}$
	Weibull–Gaussian:^e	$F (x) = {\begin{cases} 1 - \exp [- {(\frac{x}{δ})}^{m}], & x \leq x_{gr} \\ F_{gr} + \frac{r_{f}}{σ \sqrt{2 π}} \iint_{x_{gr}}^{x} \exp [- {(\frac{x^{'} - μ}{2 σ})}^{2}] d x^{'}, & x > x_{gr} \end{cases}$
	DLB:	$F (x) = 1 - \exp [- κ \cdot \exp (- δ \cdot x^{- 1 / ρ})]$

a

λ

: location;

δ

: scale; and

(ρ, a, b)

: shape parameters.

b

Duxbury–Leath–Beale (DLB) distribution (Duxbury et al. 1987).

c

The probability density function (PDF) is presented instead of the cumulative density function (CDF) because the beta distribution has no closed expression for its CDF.

d

Γ (\cdot)

: Gamma function.

e

F_{gr}

and

x_{gr}

: grafting probability and grafting point.

To ensure that

F (x)

does not exceed 1, the factor

r_{f}

needs to be scaled (Bažant and Pang 2007) as

r_{f} = \frac{1 - F_{gr}}{1 - Φ_{gr} (μ, σ)}

(2)

where

Φ_{gr} (μ, σ)

= normal distribution

N (μ, σ)

evaluated at the grafting point. Furthermore, to ensure continuity of the PDF at

x_{gr}

, the condition

f (x_{gr}^{-}) = f (x_{gr}^{+})

(3)

must hold.

A cornerstone in this theory is the representative volume element. A RVE can be defined as the smallest volume whose failure triggers the collapse of the whole structure (Pang et al. 2008), that is, the weakest link in a (finite) chain. In the present work, no effort was made to try to find the real RVE of the studied material. Instead, the 100-mm cells were assumed to work as RVEs, and their suitability was assessed by means of extreme value theory, as described subsequently. It should be clear, however, that each cell should be thought of as a hierarchical model of links connected in series and parallel. This means that local failures may occur within this RVE prior to achieving ultimate load because load sharing occurs internally.

Application of Survival Analysis to the Tensile Strength Results

Survival analysis deals with assessing what is commonly known as censored data and is extensively used in different fields of science and technology. A common case of censoring in civil and mechanical engineering arises, for example, in the duration of load tests, when, at some given time

t_{i}

, some specimens have failed, whereas others are still intact. For the survivor specimens, it is only known that their respective failure times lie above the time

t_{i}

(Nelson and Hahn 1972). For the present case of studied oak boards, the censored data correspond to the tensile strength

f_{t, 0, cell}

associated with the unbroken cells, that is, those cells characterized only by an intrinsic tensile strength that is larger than a certain value. Survival analysis has been used before in a similar study by Fink (2014); however, the methodological approach was very different. Specifically, the likelihood function (see subsequently) was not considered, and, instead, an iterative method was applied.

The problem consists of estimating the set of parameters

θ

of a candidate distribution function

F (x)

that describe the

f_{t, 0, cell}

variation. For this, maximum likelihood estimation (MLE) is commonly used, where the usual likelihood function (

L (θ)

) is modified to consider the censored data (Odell et al. 1992) according to

L (θ | X) = \prod_{i = 1}^{n} f {(x_{i})}^{δ_{i}} \cdot {[1 - F (x_{i})]}^{(1 - δ_{i})}

(4)

where

f (x)

= probability density function (PDF); and

δ_{i}

= indicator parameter signalizing whether the data were observed (censored), that is

δ_{i} = {\begin{cases} 1, & if x_{i} is an observed data point \\ 0, & if x_{i} is a censored data point \end{cases}

(5)

The main characteristic of the likelihood function [Eq. (4)] is the use of the survival function

S (x) = [1 - F (x)]

for the censored data instead of the PDF. For practical reasons, it is typical to work with the logarithm of the likelihood function, then reading

\log L (θ | X) = \sum_{i = 1}^{n} δ_{i} \cdot \log [f (x_{i})] + (1 - δ_{i}) \cdot \log [1 - F (x_{i})]

(6)

Maximizing this function yields the maximum likelihood estimates

\hat{θ}

for the tested distributions.

In the presented analysis, six different strength distribution models were fitted to the

f_{t, 0, cell}

data. First, four censored parametric models were chosen: (a) a two-parameter (2p) Weibull distribution, (b) a beta distribution, (c) a grafted Weibull–Gaussian distribution, and (d) a Duxbury–Leath–Beale (DLB) distribution (Duxbury et al. 1987). Second, two censored regression models were considered: (e) Weibull and (f) beta, where the scale (

δ

) and shape parameters (

m

, and

a

,

b

) are functions of the global MOE of each board

E_{t, 0, glob}

. The inclusion of the 2p-Weibull distribution in the analysis is bound to the general assumption in timber engineering of adopting the weakest-link theory, although it is questionable due to the quasibrittle nature of the specific material. Nevertheless, the 2p-Weibull distribution shall serve as a reference point with the models derived specifically for quasibrittle materials, that is, models (c) and (d), evidencing possible problems of applying the Weibull distribution to analyze the experimental results. The inclusion of the beta distribution is not related to any underlying theory for the distribution of strength. It was chosen for its characteristic of being bounded at both sides, thus defining a lower and upper limit, which might be a desirable feature for stochastic finite-element (FE) simulations. These six models are presented in Table 1.

The data fed to the models were constructed in the following way:

1.

A binary vector

δ

of size

15 \cdot N

(

N

= total number of tested boards; multiplier 15 represents the number of 100-mm-long cells per board) was assembled with either “1” or “0,” corresponding to whether a failure has been observed in the cell or not, respectively.

2.

A second vector of tensile strengths

f_{t, 0}

of size

15 \cdot N

is created in the following manner:

•

if

δ_{i} = 1

, then the observed

f_{t, 0, i}

value is inserted, and

•

if

δ_{i} = 0

, then the value assigned corresponds to the highest tensile strength registered during a test in which the

i

th cell was part of the free length of the board. This is interpreted as “this cell has at least a tensile strength

f_{t, 0, i}

.”

3.

For the case of models (e) and (f), representing the censored regressions, a third vector was created with the corresponding

E_{t, 0, glob}

value for each cell.

The censored analysis was implemented with the Julia programming language (Bezanson et al. 2017), in which the Optim.jl package (Mogensen and Riseth 2018) was used to minimize the log-likelihood of each model. For the case of the grafted Weibull–Gaussian model (c), the likelihood function [Eq. (6)] was modified to consider the compatibility condition at the grafting point, that is,

f (x_{gr}^{-}) = f (x_{gr}^{+})

, as

\log L (θ | X) = \sum_{i = 1}^{n} δ_{i} \cdot \log [f (x_{i})] + (1 - δ_{i}) \cdot \log [1 - F (x_{i})] - λ_{0} \cdot Δ_{gr}

(7)

where

Δ_{gr} = [f (x_{gr}^{-}) - f (x_{gr}^{+} {)]}^{2}

; and

λ_{0}

= Lagrange multiplier. Therefore, the grafting point is considered an additional parameter to be fitted.

Within the presented analysis, the data were available for two lengths: the full free length of the boards of 1,500 mm and the cells of 100 mm in length because, under the framework of the FWL theory, the Weibull exponent

m

of the grafted Weibull–Gaussian distribution should be the same for the different sizes, Eq. (7) is maximized simultaneously for both lengths, where the Weibull exponent

m

is shared by both lengths, whereas the rest of the parameters are independent for each size. In this case, the typical definition of likelihood is used for the

f_{t, 0, glob}

data, modified into a Lagrangian function with the extra term

λ_{0} \cdot Δ_{gr}

. The source code of the models is available online in Tapia and Aicher (2021a).

Model for the Simulation of Tensile Strength Profiles

The simulation of localized tensile strength values

f_{t, 0, cell}

requires the previous generation of MOE profiles along the board

E_{t, 0, cell}

, as described by Tapia and Aicher (2021b). In a very brief summary, the presented simulation of

E_{t, 0, cell}

profiles consists of three steps: (1) generation of a Gaussian

N (0, 1)

first-order autoregressive [AR(1)] process, creating normalized autocorrelated MOEs, termed

Z_{t, 0, cell}

, then (2) mapping the

Z_{t, 0, cell}

process into the so-called normalized MOE distribution, resulting in a new stochastic process termed

{\tilde{E}}_{t, 0, cell}

, and finally (3) the values

{\tilde{E}}_{t, 0, cell}

are multiplied by a scaling factor

m_{0}

to obtain the simulated

E_{t, 0, cell}

values.

The currently proposed method to simulate localized

f_{t, 0, cell}

strengths was based on the concept of a slightly modified vector autoregressive (VAR) model, in which a set of stochastic processes are cross-correlated. In this case, a Gaussian process

N (0, 1)

for normalized tensile strengths

S_{t, 0, cell}

was generated based on a cross-correlation model with the previously simulated

Z_{t, 0, cell}

values. The values

S_{t, 0, cell}

were then mapped into the correct distribution for

f_{t, 0, cell}

, estimated according to the previously described survival analysis methodology. The relevant steps for the simulation of

f_{t, 0, cell}

are as follows:

•

First, an AR(1) model for the stationary process

Z_{t, 0}

was simulated as

Z_{t, 0, i} = φ_{1} \cdot Z_{t, 0, i - 1} + ε_{i}

(8)

where

φ_{1}

= Gaussian parameter of the model; and

ε_{i}

= white noise component with a standard deviation

σ_{1} = \sqrt{1 - φ_{1}^{2}}

.

•

Second, the normalized tensile strength profile

S_{t, 0, cell}

, which depends on

Z_{t, 0, cell}

, was simulated as

S_{t, 0, i} = θ_{0} \cdot Z_{t, 0, i} + ε_{i}

(9)

where

θ_{0}

= cross-correlation coefficient between the normalized values of MOE and

f_{t, 0, cell}

. The white-noise term

ε_{i}

had, for this case, a standard deviation

σ_{2} = \sqrt{1 - θ_{0}^{2}}

. The parameters

φ_{1}

and

θ_{0}

, which have been previously determined (Tapia and Aicher 2018, 2021b), are given in Table 2.

•

Third, the vector

S_{t, 0, cell}

, which corresponds to all the

S_{t, 0, i}

values in a board and was assumed to belong to a

N (0, 1)

distribution, was mapped into the

f_{t, 0, cell}

distribution, which may be any of the fitted distributions

F (x)

presented in Table 1 by means of the transformation

f_{t, 0, cell} = F^{- 1} (Φ (S_{t, 0, cell}))

(10)

where

F^{- 1}

= inverse of the cumulative distribution function (CDF); and

Φ

= CDF of the standard normal distribution. The transformation of Eq. (10) is commonly used to translate data from one distribution into another by preserving the original CDF [see, among others, Grigoriu (1984), Taylor and Bender (1988), Chen and Gopinath (2000), and Deodatis and Micaletti (2001)].

Table 2. Parameters derived previously for the simulation model for the local tensile strength

Parameter description	Parameter	All cells	KAR < 0.05
Coefficients for the AR(1) process for $Z_{t, 0, cell}$	$φ_{1}$	0.44	0.58
Localized cross-correlation coefficient MOE versus $f_{t, 0}$	$θ_{0}$	0.75

Source: Data from Tapia and Aicher (2018, 2021b).

The vector

f_{t, 0, cell}

obtained via the outlined procedure corresponded to the simulated tensile strength profile, which has the following properties: (1) cross-correlation with

E_{t, 0, cell}

through the

θ_{0}

term, (2) implicit autocorrelation owed to the cross-correlation with

E_{t, 0, cell}

, and (3) statistical behavior equal to that obtained from the survival analysis performed with the experimental data. These characteristics, especially the last one, are analyzed subsequently in more detail.

Order Statistics and Extreme Value Theory

Order statistics addresses the statistical analysis of the

r

th smallest value in a sample of size

n

. More formally, if

X_{1}, X_{2}, \dots, X_{n}

constitute a sample drawn from a given distribution

F (x)

and are sorted in ascending order, such that

X_{1 : n} \leq \dots \leq X_{n : n}

, then the

r

th element of this sequence is defined as the

r

th-order statistic of the sample (Castillo et al. 2005). Of special interest are the first and last elements, that is, the minimum and maximum of

X_{1}, \dots, X_{n}

, respectively, generally referred to as the extreme values.

In the current study, the tensile strength is analyzed, which means that the minimum value, that is, the first-order statistic, has the highest relevance. In the considered empiric database, each board, subdivided into 15 cells of 100 mm in length, represents a sample of size 15, from which at least the minimum value (

f_{t, 0, glob}

) is known. If the parent distribution

F (x)

, that is, the distribution considering

n = 15

cells, is known, then the distribution of the minimum value

F_{\min} (x)

can be simply derived (Castillo et al. 2005) from

F (x)

as

F_{\min} (x) = 1 - {[1 - F (x)]}^{n}

(11)

Bound to a correct fitting of the parametric models (a) to (d), as shown in Table 1, the application of Eq. (11) to the

f_{t, 0, cell}

distribution must yield the distribution for

f_{t, 0, glob}

, as shown subsequently.

Results and Discussion

Observed Variation of Tensile Strength for All Boards

The results obtained from the tensile tests for the global tensile strength

f_{t, 0, glob}

are presented in Table 3. The strength results of the secondary loading

f_{t, 0, i}

are omitted from the table because they cannot be presented in a meaningful, aggregated manner due to the different free lengths of each specimen (discussed subsequently). The mean and standard deviation of

f_{t, 0, glob}

for the LS10 and LS13 grades were

27.4 \pm 10.6

and

30.5 \pm 11.3 MPa

, respectively. The characteristic values, computed according to EN 14358 (CEN 2016) for a lognormal distribution, were 12.8 and 13.9 MPa for LS10 and SL13 boards, respectively.

Table 3. Statistical evaluation of the global tensile strength values of the oak boards, separately for the different hardwood strength grades according to DIN 4074-5 (DIN 2008)

Grade	N	Mean (MPa)	Std (MPa)	$f_{t, 0, k}$ (MPa)	COV (%)	Minimum (MPa)	Maximum (MPa)
Reject	4.0	20.8	4.1	—	20.0	14.9	25.1
LS7	4.0	29.6	12.7	—	43.0	8.6	40.4
LS10	17.0	27.4	10.6	11.8	39.0	13.1	49.2
LS13	22.0	30.5	11.6	13.0	38.0	14.6	55.3
LS10+LS13	39.0	29.1	11.3	13.0	39.0	13.1	55.3
All	47.0	28.5	11.2	12.3	39.0	8.6	55.3

In a first approach to the analysis, Fig. 3(a) presents all (95) tensile strength values of those boards (

n = 37

) where multiple strength measurements were possible. These data were obtained either in the first loading to fracture

f_{t, 0, glob}

or in the secondary loading tests

f_{t, 0, \sec}

. Fig. 3(b) presents the same results, now as ratios

f_{t, 0, \sec} / f_{t, 0, glob}

. The chosen presentation of the data enables a first assessment of the variability of

f_{t, 0, cell}

in the studied oak boards.

Fig. 3. Tensile strengths $f_{t, 0}$ for boards with more than one tensile strength measurement: (a) all strengths per board; and (b) ratios of secondary versus global strengths.

In both Figs. 3(a and b), the boards are ordered from left to right by the assigned hardwood strength grade (LS) and within each LS grade group by ascending strength ratio

f_{t, 0, \sec} / f_{t, 0, glob}

within each board. Substantial variations denoted by strength ratios up to almost 6 can be observed. However, the mean and standard deviation of the regarded strength ratio of 2.2±1.2 fosters the impression that this variation usually is not that extreme. Nevertheless, such an assessment is biased because the data are highly incomplete, owing to the physical impossibility of testing the tensile strength of each cell. As such, the real variation should probably be higher. This is analyzed subsequently in more depth.

Estimated Parameters of the Local Tensile Strength Models

Table 4 presents the estimated parameters of the six studied strength distribution models, which were fitted to the tensile strength data

f_{t, 0, cell}

of the whole set (

N = 47

) and to the subset of grades LS10 and LS13 boards (

N = 39

). Further, the

\log L

and the Akaike information criterion (AIC) (Akaike 1974) are given for each model. The AIC serves to discriminate between different models based on the likelihood and number of parameters (

AIC = 2 k - 2 \log L

, with

k

parameters). For the computation of the AIC of the Weibull–Gaussian (c) model,

k = 4

because the model is fully defined by the parameters

δ

,

m

,

μ

, and

σ

, whereas

x_{gr}

is only needed for the fitting process. Due to numerical problems, the beta regression model could not be fitted to the data subset of LS10 and LS13 boards.

Table 4. Parameters and statistical indicators for

f_{t, 0, cell}

estimated for the different fitted distributions

Model		Loc	Scale	Shape		$\log L$	AIC
Model		$λ$ , $μ$	$δ$	$m$ , a, $κ$	b, $σ$ , $ρ$	$\log L$	AIC
All boards:
(a)	Weibull	—	$7.72 \times 10^{1}$	2.90	—	$- 601.5$	1,207.1
(b)	Beta	6.93	$9.34 \times 10^{1}$	2.15	1.23	$- 600.3$	1,208.5
(c)	Wei.-Gauss.	67.9	$3.58 \times 10^{1}$	5.80	$2.80 \times 10^{1}$	$- 599.8$	1,207.6
(d)	DLB	—	$2.96 \times 10^{1}$	$1.19 \times 10^{5}$	4.73	$- 601.1$	1,208.1
(e)	Weibull (reg.)	0	$α_{0}$ : 4.04	$β_{0}$ : $2.22 \times 10^{- 1}$	—	$- 591.8$	1,191.5
(e)	Weibull (reg.)	0	$α_{1}$ : $2.36 \times 10^{- 5}$	$β_{1}$ : $8.01 \times 10^{- 5}$	—	$- 591.8$	1,191.5
(f)	Beta (reg.)	0	$α_{0}$ : 5.90	$β_{0}$ : $2.02 \times 10^{- 1}$	$γ_{0}$ : 2.01	$- 590.5$	1,193.1
(f)	Beta (reg.)	0	$α_{1}$ : $5.50 \times 10^{- 5}$	$β_{1}$ : $1.19 \times 10^{- 4}$	$γ_{1}$ : $1.48 \times 10^{- 4}$	$- 590.5$	1,193.1
$LS 10 + LS 13$ grades:
(a)	Weibull	—	$7.39 \times 10^{1}$	3.15	—	$- 492.4$	988.7
(b)	Beta	10.6	$7.93 \times 10^{1}$	1.97	1.00	$- 490.1$	988.1
(c)	Wei.-Gauss.	65.8	$2.72 \times 10^{1}$	9.32	$2.52 \times 10^{1}$	$- 489.6$	987.2
(d)	DLB	—	$3.17 \times 10^{1}$	$2.73 \times 10^{5}$	4.68	$- 492.3$	990.6
(e)	Weibull (reg.)	0	$α_{0}$ : 3.91	$β_{0}$ : $4.84 \times 10^{- 1}$	—	$- 484.9$	977.9
(e)	Weibull (reg.)	0	$α_{1}$ : $3.11 \times 10^{- 5}$	$β_{1}$ : $6.34 \times 10^{- 5}$	—	$- 484.9$	977.9

Note: reg. = regression; Wei.–Gauss. = Weibull–Gaussian; Loc = location; DLB = Duxbury–Leath–Beale; and AIC = Akaike information criterion.

According to the AIC, the Weibull regression model (e) is the best of the six models to simulate the censored

f_{t, 0, cell}

data (smallest AIC). This means that the incorporation of the modulus of elasticity data

E_{t, 0, glob}

has a positive and meaningful impact on the description of

f_{t, 0, cell}

. This observation applies to the subgroup of LS10+LS13 grades, too. Considering only the parametric models, the Weibull–Gaussian (c) presents the maximum

\log L

value, meaning that the quality of the fitting is better for this model, with the beta (b) model also being relatively close.

Figs. 4 and 5 show the CDF and PDF distributions of localized tensile strength according to all four fitted parametric models, for all boards and the subsample of LS10+LS13 boards, respectively. The CDF distributions in Figs. 4(a) and 5(a) are given in a Weibull plot representation and compared to the nonparametric Kaplan–Meier (KM) estimator (Kaplan and Meier 1958).

Fig. 4. Distributions of localized tensile strength $f_{t, 0, cell}$ from censored parametric Weibull, Weibull–Gaussian, beta, and DLB models (all boards and grades): (a) CDF including Kaplan–Meier estimator curve; and (b) PDF. Shaded area corresponds to the [5%, 95%] confidence interval of KM estimator.

Fig. 5. Distributions of localized tensile strength $f_{t, 0, cell}$ from censored parametric Weibull, Weibull–Gaussian, beta, and DLB models (grades LS10+LS13): (a) CDF including Kaplan–Meier estimator curve; and (b) PDF. Shaded area corresponds to the [5%, 95%] confidence interval of KM estimator.

Good agreement can be observed between all model curves and the KM estimator on the upper part of the strength distribution. However, as apparent, the models behaved fundamentally differently at the lower tail of the distributions. As predicted by the FWL theory, a rather abrupt change in the slope of the data was observed—most pronounced in the LS10+LS13 subgroup [Fig. 5(a)]. It is clear that the 2p-Weibull distribution could not follow this curve because the equation represents a straight line in the Weibull plot. The beta distribution seems to fit the data much better. However, because it imposes a lower limit

λ = 6.9 MPa

and 10.6 MPa for the case of all boards and the

LS 10 + LS 13

subgroup, respectively, this model is expected to present similar problems as the 3p-Weibull distribution [see Bažant and Le (2017), and Pang et al. (2008)], which is explained further subsequently. The WG model (c) presents a fit that is visually similar to the beta model, with matching likelihood values, too. The grafting point—marked with a circle in Figs. 4(a) and 5(a)—was determined as

x_{gr} = 15.0 MPa

and 14.6 MPa for all boards and the LS10+LS13 subgroup, respectively. Finally, the DLB distribution, although in principle able to follow the specific nonlinear type of curve, could not be fitted to the same degree as, for example, the WG distribution (c), which is graphically and numerically evident from Figs. 4(a) and 5(a) and Table 4.

The differences on the midupper region of the distributions are evidenced in Figs. 4(b) and 5(b), where models (a), (c), and (d) show similar behaviors, whereas model (b), beta, is characterized by a sudden drop in its PDF, marking the end of its support. This drop was estimated for the case of all boards [Fig. 4(a)] as

loc + scale \approx 100 MPa

. For the case of the LS10+LS13 subgroup, the beta model (b) exhibited a sharp end at about 90 MPa, determined by the shape parameter

b = 1

[Fig. 5(a)]. The beta model was the only model presenting a clear skewness “to the left,” which is expressed by an abrupt descent of the right tail, whereas all other models showed more symmetric behaviors, even slightly right-skewed, reaching higher

f_{t, 0, cell}

values.

Figs. 6(a and b) illustrate the results of the fitted Weibull and beta regression models (e) and (f) for all grades, respectively, by means of several PDF curves corresponding to different

E_{t, 0, glob}

values in the range of 8–16 GPa. The expected mean strength values of the regression models fitted to the

LS 10 + LS 13

subset are shown in Fig. 6(a), too. Here, a clear increase in the expected value of

f_{t, 0, cell}

was observed from both models with growing

E_{t, 0, glob}

. Thus, these two models capture both the variation of

f_{t, 0, cell}

and its correlation with the global MOE of boards. The behavior of both models was, however, different. The Weibull model (e) started with a rather right-skewed (

s = 0.42

) and widespread distribution for lower

E_{t, 0, glob}

and evolved toward a more symmetric distribution (

s = - 0.18

) with less spread for higher

E_{t, 0, glob}

values. The beta regression model (f) started with a widespread, right-skewed distribution for lower

E_{t, 0, glob}

values, too. However, in contrast to the Weibull model, the shape of the PDFs stayed rather constant for higher

E_{t, 0, glob}

, only shifted upward toward higher

f_{t, 0, cell}

values.

Models also considering the location parameter

λ

as a free parameter (i.e.,

λ \neq 0

, both dependent and independent of

E_{t, 0, glob}

) did not render satisfactory results but, in fact, resulted in mechanically unlikely distributions. This could be related to insufficient data points or numerical instabilities, preventing finding the optimum result. For highly nonlinear problems with many parameters to fit, choosing the right initial values has an important influence on the outcome. For the analyzed models with

λ = 0

, the estimated parameters represented the optimal ones, which was proven by checking multiple different combinations of the initial conditions.

Verification of the Parametric Models by Extreme Value Theory

The goodness of the fitted parametric beta (b) and Weibull–Gaussian (c) models can be checked by extreme value theory. As previously mentioned, if the fitted models describe the tensile strength variation along the board, then the application of Eq. (11) to the CDF of the fitted models should match the experimental values of

f_{t, 0, glob}

(minimum value of each board). This is demonstrated graphically in Figs. 7 and 8 for all boards and for the LS10+LS13 subsample, respectively. The experimental results were plotted by applying the midpoint position method

P (x \leq x_{i}) = (i - 0.5) / n

(Rinne 2009).

Fig. 7. Global tensile strength derived from parametric models for localized strength of all boards using order statistics [Eq. (11)]: (a) Weibull–Gaussian; and (b) beta.

Fig. 8. Global tensile strength derived from parametric models for localized strength of subgroup $LS 10 + LS 13$ using order statistics [Eq. (11)]: (a) Weibull–Gaussian; and (b) beta.

It is apparent that the theoretical minimum CDF curve

F_{\min} (x)

associated with model (c) matched rather exactly both the experimental results and the WG distribution fitted to the global results (fitted simultaneously with local values, as described previously) using an exponent

n = 15

in Eq. (11). This is the desired behavior in a stochastic model for

f_{t, 0, cell}

because this means that the model should be able to capture the size effect (discussed subsequently) in a relatively good manner. It further ensures that a power law describes the size effect of the lower tail. The behavior of model (b), illustrated in Figs. 7(b) and 8(b), showed in general good agreement with the global data. Nevertheless, the observed behavior at the lower tail indicates that the size effect will not be governed by a power law, which is the same problem as observed with the 3p-Weibull distribution; that is, the model will tend to an asymptotic limit for larger lengths (Bažant and Le 2017). However, for realistic board lengths of the regarded hardwood species, limited to a few meters (

max. \approx 3 m

), the problem might not be completely evident from simulation results (see subsequently).

Regarding the censored regression models (e) and (f), a similar analysis could not be performed because their parameters depend on

E_{t, 0, glob}

. This would need a comparison against experimental data for different specific

E_{t, 0, glob}

values, which is not possible for this data set due to the rather small sample size.

Simulation of Tensile Strength Profiles along Board Length

Based on the estimated parameters of the studied models for

f_{t, 0, cell}

, tensile strength profiles of virtual boards can be generated using Eqs. (8)–(10) of the described vector autoregressive model. Figs. 9(a and b) show profiles of

f_{t, 0, cell}

along two boards (I, II) generated with the beta model (b). Furthermore, the figures include the corresponding

E_{t, 0, cell}

values, simulated according to Tapia and Aicher (2021b).

Fig. 9. Simulated tensile strength profiles along the length of two virtual boards, I and II, with the fitted parametric beta model, based on cross-correlated MOE distributions: (a) Board I; and (b) Board II.

For both simulated profiles, a clear correlation between

E_{t, 0, cell}

and

f_{t, 0, cell}

can be observed. Both curves moved similarly, but differences arose due to the introduced white noise component

ε_{i}

in Eq. (9). In general, regions of lower

E_{t, 0, cell}

values also present relatively low

f_{t, 0, cell}

values, and vice versa. The

E_{t, 0, glob}

value needed for MOE profile simulation was taken randomly from the statistical distribution for global MOE given in Tapia and Aicher (2021b).

The relatively high variation in both

E_{t, 0, cell}

and

f_{t, 0, cell}

was due to the characteristics of the studied oak board material, which made up in essence two strength grades, LS10 confined at the lower and upper bounds by grades LS7 and LS13, respectively, and LS13 and better. Nevertheless, the presented method is independent of the specific board material and considered applicable to different data sets showing more or less internal MOE or tensile strength variation.

Simulation of Length Effect for Tensile Strength

The decrease of strength with the growing size of wooden (compound) materials was first addressed by Newlin and Trayer (1924) and Freas and Selbo (1954) in a phenomenological manner and related to bending strength as dependent on cross-sectional depth. In addition, Weibull (1951) derived a universal mechanical–mathematical model for the effect of size on strength of brittle materials, based on a higher probability of occurrence of a more detrimental defect, also termed weakest link (WL) in a larger structure. Subsequently, Barrett (1974) introduced the (WL) approach for characterizing quasibrittle material wood, and then, specifically, the size effect in tension perpendicular to the fiber direction. First investigations on the effect of length of boards on tensile strength parallel to fiber—all related (implicitly) to the WL concept—were reported by Showalter et al. (1987), Bechtel (1988), Lam and Varoglu (1990), Barrett and Fewell (1990), Madsen (1990), Rouger and Barrett (1995), Burger and Glos (1996), and Zhou et al. (2010) and, more recently, for example, by Fink (2014) and Blank et al. (2017). Findings of some of the mentioned studies are discussed subsequently in comparison with the results derived here, including the finite weakest-link theory (Bažant and Pang 2007), which so far has not been applied for the description and simulation of length effects in timber.

The fact that the previously presented modified vector autoregressive model can simulate the effect of size, that is, of board length, on the tensile strength results directly from the “cell”-centered stochastic nature of the model and is tightly related to its compliance with the extreme value theory (shown previously). In order to demonstrate the size effect produced by the simulation method, virtual boards of seven different lengths, 500–3,500 mm in steps of 500 mm, were generated applying the fitted models (b), (c), (e), and (f) for

f_{t, 0, cell}

considering all grades and the subgroup LS10+LS13. For each length, a total of 50,000 simulations per

f_{t, 0, cell}

model were performed. The simulation of different lengths does not require any special consideration beyond generating the needed amount of cells according to Eqs. (8)–(10). After this, the smallest

f_{t, 0, cell}

value of each virtual board was taken as the

f_{t, 0, glob}

value.

Fig. 10 shows the results of the simulations for the seven different board lengths, based on the parameters specified in Table 4 for all grades. A pronounced size effect can be observed for the four studied models, with diminishing strength values as the length increases. Table 5 gives the means and standard deviations of the tensile strength of the different board lengths for the fitted

f_{t, 0, cell}

models. It is evident from Table 5 that the parametric beta (b) and Weibull–Gaussian (c) models show a very similar evolution in both mean strength values and spread. More importantly, regarding the agreement with the empiric values (Table 3), both models simulate the mean tensile strength of the reference length (

n = 15 = 1500 mm

) almost exactly. Similarly, the standard deviation is just slightly underestimated by 4% for models (b) and (c).

Fig. 10. Simulated length effect of tensile strength obtained with the four fitted models [(b) beta, (c) Weibull–Gaussian, (e) Weibull regression, and (f) beta regression] for the $f_{t, 0, cell}$ variation of all grades. Outer lines indicate 5% and 95% quantiles.

Table 5. Simulated global tensile strength values

f_{t, 0, glob}

(

mean \pm std.

), all grades, for models (b), (c), (e), and (f) and different board lengths

$l_{board}$ (mm)	$Mean \pm std.$ (MPa)
$l_{board}$ (mm)	500	1,000	1,500	2,000	2,500	3,000	3,500
Model
(b)	$41.8 \pm 16.7$	$32.7 \pm 12.9$	$28.3 \pm 10.8$	$25.6 \pm 9.4$	$23.7 \pm 8.5$	$22.3 \pm 7.8$	$21.2 \pm 7.3$
(c)	$41.6 \pm 16.8$	$32.5 \pm 12.8$	$28.1 \pm 10.7$	$25.4 \pm 9.3$	$23.5 \pm 8.4$	$22.2 \pm 7.6$	$21.1 \pm 7.1$
(e)	$42.3 \pm 16.4$	$33.9 \pm 13.6$	$29.8 \pm 12.2$	$27.1 \pm 11.3$	$25.2 \pm 10.7$	$23.8 \pm 10.2$	$22.7 \pm 9.8$
(f)	$42.6 \pm 16.9$	$34.1 \pm 13.1$	$30.1 \pm 11.5$	$27.7 \pm 10.5$	$26.0 \pm 9.9$	$24.7 \pm 9.5$	$23.6 \pm {9.11}^{''}$

Regarding the results of the regression-based

f_{t, 0, cell}

models, the Weibull model (e) showed a similar size effect trend as those obtained for models (b) and (c). However, the mean and standard deviation for the reference length deviated from the empiric results by 5% and 9%, respectively. Similar behavior was observed for model (d), in which the mean and standard deviation differed from the experiments by 6% and 3%, respectively. The reason for the slightly higher deviations of the regression models is probably related to the specific

E_{t, 0, glob}

distribution used to generate the

E_{t, 0, glob}

input data because the models directly depend on it. A larger data set would probably help improve the model prediction quality because both the parameters estimated for the distribution of

E_{t, 0, glob}

and models (e) and (f) would be determined with higher confidence.

To quantify the simulated length effect comprehensively, a power law of the form

f_{t, i} = f_{t, ref} \cdot {(\frac{l_{ref}}{l_{i}})}^{ξ}

(12)

was fitted to the results of each of the four models, where

f_{t, i}

= tensile strength for a length

l_{i}

; and the reference length

l_{ref} = 1,500 mm

. The quantification of the size effect by means of a power law as a generalized description of the size impact, irrespective of underlying failure mechanisms (WL-concept or fracture mechanics), has become an internationally adopted approach in test evaluation and timber design codes, for example, EN 384 (CEN 2018), EN 1995-1-1 (CEN 2014), ASTM D245 (ASTM 2019), and ASTM D6570-18a (ASTM 2018). The obtained results for the exponent

ξ

are presented in Table 6 and Fig. 11. Figs. 11(a and b) illustrate the mean and 5%-quantile results for the simulated data. Disregarding first the differences between the slopes at the mean and 5%-quantile levels, a clear linear trend was observed in the log-log plot of the

f_{t, 0}

values versus length for all cases. This implies that the assumed power law of Eq. (12) is adequate for the investigated lengths at both distribution levels. Considering all boards, the size effect exponents

ξ

obtained on the mean level ranged from 0.31 to 0.35, whereas for the 5% quantile,

ξ

moved between 0.18 and 0.34. For the subset of grades LS10 and LS13 only, the exponents

ξ

were slightly lower, ranging from 0.30 to 0.33 on the mean level and between 0.14 and 0.31 for the 5% quantile.

Table 6. Size exponents

ξ

describing the length effect of the tensile strength for the sets of boards simulated by models (b), (c), (e), and (f)

$ξ$ -level	Model
$ξ$ -level	(b)	(c)	(e)	(f)
All grades
$ξ$ –mean	0.35	0.35	0.32	0.31
$ξ$ –5%-quant.	0.22	0.17	0.34	0.28
$LS 10 + LS 13$ grades:
$ξ$ –mean	0.32	0.32	0.30	—
$ξ$ –5%-quant.	0.19	0.14	0.31	—

Note: quant. = quantile.

Fig. 11. Assessment of power law assumption for strength results of boards simulated with different models [(b), (c), (e), and (f)] at two distribution levels: (a) mean; and (b) 5%-quantile.

The differences observed for the

ξ

values between the mean and 5%-quantile levels are best explained in terms of FWL theory. FWL predicts that a power law such as that described in Eq. (12) is only valid in the asymptotic limit, that is, for large enough structures in which the Weibull part of the Weibull–Gaussian distribution clearly dominates. In this limit, the relation

ξ = 1 / m

must hold, where

m

is the Weibull module of the fitted WG distribution. It is immediately evident that the 5%-quantile level has already reached or is very close to the asymptotic limit, given that for all boards,

ξ = 0.17 = 1 / 5.9 \approx 1 / 5.5 = 1 / m

, and for the LS10+LS13 subgroup,

ξ = 0.14 = 1 / 7.1 \approx 1 / 9.0 = 1 / m

.

The general expression for the size effect of quasibrittle materials as proposed by Bažant et al. (2009), Le et al. (2011), and Carloni et al. (2019), is defined as

σ_{N c} = f_{r_{\infty}} {[{(\frac{D_{a}}{D})}^{r n / m} + \frac{r D_{b}}{D}]}^{1 / r}

(13)

where

σ_{N c}

= global strength;

f_{r_{\infty}}

= strength parameter related to the fracture process zone and material energy;

D_{a}

,

D_{b}

, and

r

= scaling parameters;

m

= Weibull modulus; and

n

= number of spatial dimensions in which the structure is scaled. Eq. (13) describes a monotonically decreasing function with an asymptotically decreasing slope that converges to 1/m (Fig. 12). This relation has been experimentally verified for different quasibrittle materials [e.g., Le et al. (2013)]. It can be deduced, then, that the mean values are still in the transition zone of the curve defined by Eq. (13) because the sizes are not large enough to make the Weibull component of the WG relevant. However, it is reasonable to think that this is not necessarily true for the 5%-quantile level, given that there is a higher probability for these values to be bound to the lower tail of the distribution, that is, to the Weibull region of the WG distribution. For example, the parameter

D_{b}

in Eq. (13) may be different at the mean and 5%-quantile level, which is illustrated in Fig. 12 for arbitrary parameters. In this manner, the observed difference between the size effects computed for the mean and 5%-quantile levels is coherently explained, and the observed difference in

ξ

is only a consequence of the quasibrittle nature of the studied material and the simplified scaling law of Eq. (12).

Fig. 12. Theorized relationship between size effect on the mean and 5%-quantile levels.

Regarding an assessment of the derived exponents for oak boards and any other hardwood species, no reference values could be found. Hence, in a first approach, the quantitative discussion of the simulation-based

ξ

values is related to the size factors given in the literature for softwoods.

Based on a metastudy of an extensive database (27 subsets, 3,070 specimens) by Showalter et al. (1987), Lam and Varoglu (1990), and (Madsen 1990) on Canadian spruce–pine–fir (SPF) and Southern pine (US) of different structural grades, Barrett and Fewell (1990) proposed an exponent of

ξ = 0.17

for the 5%-quantile level [5%-quantiles when computed according to CEN TC 124.202 (CEN 1989)]. Further, it was concluded that the length effect in tension and bending is alike. Rouger and Barrett (1995) supported the stated length effect exponent

ξ = 0.17

and discussed the variables that can influence the sample size effect. These are mainly: (1) the sawing pattern; and (2) the used grading system. Specific sawing patterns may lead to an apparent size factor up to 0.37. The effect of the grading system was illustrated with lower and higher quality samples graded both visually and with regard to MOE by machine grading. Whereas for the MOE-graded material,

ξ = 0.23

was found, the visually graded boards showed

ξ

values ranging from –0.5 to 0.1. Burger and Glos (1996) conducted an experimental study with 750 European spruce (Picea abies) boards with knot area ratios ranging from 0.16 to 0.61. The valid (

n = 730

) test results from three significantly different lengths (0.15, 1.0, and 2.5 m) yielded size exponents of 0.13 and 0.22 for the mean and characteristic levels, respectively (5%-quantiles where computed by using a 3p-Weibull distribution).

Considering the cited literature on softwood results, it can be stated that the simulation-based length effect for the oak boards is well within the experimental range at the 5%-quantile level. Focusing exclusively on the subset LS10+LS13, being more relevant for structural purposes, the parametric models (b) and (c) yielded

ξ

values of 0.19 and 0.14, rather similar to the softwood results. The Weibull regression model (e) gave a much higher value, relatively close to the mean level results. Nevertheless, the computation of experimental 5%-quantiles is complicated because it is highly sensitive to the number of specimens and to the assumed strength distribution, which will impact the computed size effect. For example, the fact that the size effect exponents at the mean level are smaller than at the 5%-quantile level in Burger and Glos (1996) goes against the predictions made here based on FWL theory. Therefore, although good agreement is obtained, caution is needed when comparing simulations with experimental results at this quantile level. At the more stable mean level, the obtained results for the size exponent

ξ

are up to two times higher as compared to the literature results for softwoods. This might be due to intrinsic differences between oak and softwood species; however, additional experiments are needed to enable a decisive statement on this aspect.

From an overall perspective, it can be concluded that the presented order statistics approach, based on localized (cell-wise) strength values, provides conceivable results for the length effect. Nevertheless, the magnitude of the length effect exponent, including the model validation, must be investigated experimentally with larger samples.

Discussion

The two approaches used to estimate the within-board variation of tensile strength present some implications beyond simply simulating a

f_{t, 0}

profile for a single board. One of the major differences between the simple parametric models (a), (b), (c), and (d) and the censored regression models (e) and (f) consists in their practical application to simulate a large data set of virtual boards. As is widely known, there is a positive correlation between global MOE and tensile strength of wooden boards, which has been repeatedly shown for different species, including oak. Consequently, an aim should be to replicate such a correlation when simulating a population of boards with the presented models. In this sense, it is evident that models (e) and (f) intrinsically carry the information needed to simulate the correlation with the global MOE because

E_{t, 0, glob}

is used as a variable in the model. The parametric models, however, lack any correlation with global MOE because they correlate only to the localized

E_{t, 0, cell}

through the variable

Z_{z, 0, cell}

and would require an extra step to account for the observed

E_{t, 0} - f_{t, 0}

, correlation.

However, there is also an advantage in this latter approach; that is, by decoupling the simulation of

E_{t, 0}

and

f_{t, 0}

at a board level, it is easier to generate data that follow exactly the experimental distributions of

E_{t, 0}

and

f_{t, 0}

. This was shown in the previous section, where models (e) and (f) could not reproduce the experimental distribution for

f_{t, 0, cell}

as precisely as models (b) and (c). Of course, this is not only model-bound but probably related, as mentioned previously, to the influence of the size of the data set.

It can be argued that a model of type (b) resembles the concept of weak and strong sections used by Isaksson (1999), Fink (2014), and Blank et al. (2017), among others. This results from the fact that the fitted beta distribution will, in general, assign very similar high values to a large portion of a board (

\approx

strong sections) while producing a smaller number of regions with relatively low tensile strength (weak sections). The fact that this distribution is not compatible with the FWL theory raises a question about its applicability for simulation purposes. However, as shown, at least for the relevant board lengths, the results of the beta distribution are comparable to the Weibull–Gaussian approach.

Finally, the presented analysis demonstrates the benefit of incorporating the finite weakest link theory to the strength analysis of timber elements for brittle failure modes. This includes tensile strength parallel and perpendicular to the fiber, as well as shear strength. The fitting of the grafted Weibull–Gaussian distribution to tensile strength data of boards will yield the asymptotic size effect coefficient

1 / m

, which was shown to be relevant for the 5%-quantile level of boards, but should also be relevant for the mean level of larger structures composed of individual boards such as glued laminated timber beams. This dependency should be studied in depth because it could have a large impact on the amount and dimensions of the typically required specimens needed to assess the size effect of GLT beams.

Conclusions

An analysis of the tensile strength variation along oak boards was presented. Using methods from survival analysis, the parameters for parametric and regression models (Weibull, beta, Weibull–Gaussian, and Duxbury–Leath–Beale) to describe the strength distribution were derived. Based hereon, a simulation method for the strength variation was developed. Regarding the parametric models, the grafted Weibull–Gaussian distribution represents the variation along the board better because it aligns with the recently developed finite weakest link theory for quasibrittle materials, thus explaining correctly the size effect. Regression models in which the parameters of 2p-Weibull and beta distributions depend on the global MOE of the board fit the data better than the parametric models when judged by a pure statistical criterion (AIC indicator). However, the simulation results of different lengths showed poor performance as compared to the Weibull–Gaussian distribution because the mean and standard deviation were not correctly reproduced at the reference length of 1,500 mm.

The presented vector autoregressive model for the simulation of

f_{t, 0, cell}

profiles along boards, which includes the cross-correlation with the localized MOE profiles, produced reasonable results. Furthermore, the model allowed for a direct simulation of the size effect, which, for the case of the Weibull–Gaussian model, can be directly compared against the fitted Weibull module

m

. The simulated size effect exponent for the mean value was roughly two times higher as compared to the 5%-quantile level, the latter of which showed good agreement with experimental values given in literature for softwoods. The difference between the size exponent computed at the mean and 5%-quantile levels is explained by the finite weakest link theory by considering the general size dependency relation derived by Bažant et al. (2009). Further steps need to be taken in order to study the strength of hardwood and softwood timber elements within the framework of the finite weakest link theory because this has the potential to significantly impact the size-related design of large glued laminated timber beams. The presented model to simulate localized tensile strength profiles, together with the model for the simulation of localized MOE values presented by Tapia and Aicher (2021b), provides a substantial basis for a stochastic GLT strength model to predict the bending properties of glued laminated timber beams made of oak.

Notation

The following symbols are used in this paper:

$a$: shape parameter of beta distribution;
$b$: shape parameter of beta distribution;
$E_{t, 0, cell}$: cell-wise MOE;
$E_{t, 0, glob}$: global MOE of board;
${\tilde{E}}_{t, 0, cell}$: cell-wise normalized MOE;
$F$: cumulative distribution function;
$F^{- 1}$: inverse of cumulative distribution function;
$F_{\min}$: cumulative distribution function of minimum value;
$f$: density distribution function;
$f_{t, 0, cell}$: tensile strength of one cell;
$f_{t, 0, glob}$: global tensile strength of board;
$f_{t, 0, i}$: tensile strength of cell $i$ ;
$f_{t, 0, i}$: tensile strength of board with free length $i$ ;
$f_{t, 0, \sec}$: tensile strength obtained from secondary tensile tests;
$k$: number of parameters of model;
$L$: likelihood function;
$l$: length of board;
$l_{E}$: length used to measure global MOE;
$l_{i}$: length of board $i$ ;
$l_{ref}$: reference length of board for length effect;
$l_{s}$: free span length for global tensile tests;
$l_{s, 2, i}$: free span length for secondary tensile tests;
$m$: shape parameter of Weibull distribution;
$S_{t, 0, cell}$: stationary process for localized tensile strength;
$s$: skewness;
$t$: thickness of cross-section of board;
$w$: width of cross-section of board;
$Z_{t, 0, cell}$: stationary autocorrelation (AR) process;
$α_{0}$: intercept term for scale parameter $δ$ (listed subsequently);
$α_{1}$: regression coefficient associated with $E_{t, 0, glob}$ for parameter $δ$ ;
$β_{0}$: intercept term for shape parameters $m$ and $a$ ;
$β_{1}$: regression coefficient for parameters $m$ and $a$ associated with $E_{t, 0, glob}$ ;
$γ_{0}$: intercept term for parameter $b$ ;
$γ_{1}$: regression coefficient associated with $E_{t, 0, glob}$ for parameter $b$ ;
$δ$: scale parameter of Weibull, beta, and Duxbury–Leath–Beale distribution;
$δ_{i}$: binary variable indicating whether data are censored;
$ε_{i}$: white noise component of first-order autoregressive process;
$θ$: parameters of statistical distributions;
$θ_{0}$: cross-correlation term between cell-wise modulus of elasticity and tensile strength;
$\hat{θ}$: estimated parameters of statistical distributions;
$κ$: shape parameter of DLB distribution;
$λ$: location parameter of beta and Weibull distributions;
$λ_{0}$: Lagrange multiplier;
$μ$: location parameter of normal distribution;
$ξ$: length effect exponent;
$ρ$: shape parameter of DLB distribution;
$σ_{1}$: standard deviation of white noise term of AR(1) process;
$σ_{2}$: standard deviation of white noise term of modified VAR process;
$Φ$: cumulative distribution function of standard normal distribution $N (0, 1)$ ; and
$φ_{1}$: model parameter for AR(1) process.

Data Availability Statement

Some or all data, models, or code generated or used during the study are available in a repository or online in accordance with funder data retention policies [doi: 10.18419/darus-864, Tapia and Aicher (2021a)].

Acknowledgments

The financial support of the work by the German foundation FNR, Fachagentur Nachwachsende Rohstoffe e.V., contract 2200414, in conjunction with the European ERA-WoodWisdom project “European hardwoods for the building sector (EU Hardwoods)” is gratefully acknowledged. Many thanks are indebted to all project partners, here in particular to FCBA, Bordeaux, France.

References

Akaike, H. 1974. “A new look at the statistical model identification.” IEEE Trans. Autom. Control 19 (6): 716–723. https://doi.org/10.1109/TAC.1974.1100705.

Google Scholar

ASTM. 2018. Standard practice for assigning allowable properties for mechanically graded lumber. ASTM D6570-18a. West Conshohocken, PA: ASTM.

Google Scholar

ASTM. 2019. Standard practice for establishing structural grades and related allowable properties for visually graded lumber. ASTM D245. West Conshohocken, PA: ASTM.

Google Scholar

Barrett, J. D. 1974. “Effect of size on tension perpendicular-to-grain strength of Douglas-fir.” Wood Fiber Sci. 6 (2): 126–143.

Google Scholar

Barrett, J. D., and A. R. Fewell. 1990. “Size factors for the bending and tension strength of structural timber.” In Proc., Int. Council for Building Research Studies and Documentation. Working Comission W18A—Timber Structures, Paper CIB-WI8A/23-10-3 Meeting 23. Karlsruhe, Germany: Univ. of Karlsruhe.

Google Scholar

Bažant, Z. P., and J.-L. Le. 2017. Probabilistic mechanics of quasibrittle structures: Strength, lifetime, and size effect. Cambridge, UK: Cambridge University Press.

Crossref

Google Scholar

Bažant, Z. P., J.-L. Le, and M. Z. Bažant. 2009. “Scaling of strength and lifetime probability distributions of quasibrittle structures based on atomistic fracture mechanics.” Proc. Natl. Acad. Sci. 106 (28): 11484–11489. https://doi.org/10.1073/pnas.0904797106.

Google Scholar

Bažant, Z. P., and S. D. Pang. 2006. “Mechanics-based statistics of failure risk of quasibrittle structures and size effect on safety factors.” Proc. Natl. Acad. Sci. 103 (25): 9434–9439. https://doi.org/10.1073/pnas.0602684103.

Google Scholar

Bažant, Z. P., and S.-D. Pang. 2007. “Activation energy based extreme value statistics and size effect in brittle and quasibrittle fracture.” J. Mech. Phys. Solids 55 (1): 91–131. https://doi.org/10.1016/j.jmps.2006.05.007.

Google Scholar

Bechtel, F. K. 1988. “A model to account for length effect in the tensile strength of lumber.” In Vol. 1 of Proc., Int. Conf. on Timber Engineering, 355–361. Pullman, WA: Washington State Univ.

Google Scholar

Bertalan, Z., A. Shekhawat, J. P. Sethna, and S. Zapperi. 2014. “Fracture strength: Stress concentration, extreme value statistics, and the fate of the Weibull distribution.” Phys. Rev. Appl. 2 (3): 034008. https://doi.org/10.1103/PhysRevApplied.2.034008.

Google Scholar

Bezanson, J., A. Edelman, S. Karpinski, and V. B. Shah. 2017. “Julia: A fresh approach to numerical computing.” SIAM Rev. 59 (1): 65–98. https://doi.org/10.1137/141000671.

Google Scholar

Blank, L., G. Fink, R. Jockwer, and A. Frangi. 2017. “Quasi-brittle fracture and size effect of glued laminated timber beams.” Eur. J. Wood Wood Prod. 75 (5): 667–681. https://doi.org/10.1007/s00107-017-1156-0.

Google Scholar

Burger, N., and P. Glos. 1996. “Effect of size on tensile strength of timber.” [In German.] Holz Roh Werkst. 54 (5): 333–340. https://doi.org/10.1007/s001070050197.

Google Scholar

Carloni, C., G. Cusatis, M. Salviato, J.-L. Le, C. G. Hoover, and Z. P. Bažant. 2019. “Critical comparison of the boundary effect model with cohesive crack model and size effect law.” Eng. Fract. Mech. 215 (Jun): 193–210. https://doi.org/10.1016/j.engfracmech.2019.04.036.

Google Scholar

Castillo, E., A. S. Hadi, N. Balakrishnan, and J. M. Sarabia. 2005. Extreme value and related models with applications in engineering and science. Hoboken, NJ: Wiley.

Google Scholar

CEN (European Committee for Standardization). 1989. Structural timber: The determination of characteristic values of mechanical properties and density of timber. CEN TC 124.202. Brussels, Belgium: CEN.

Google Scholar

CEN (European Committee for Standardization). 2009. Sawn timber—Appearance grading of hardwoods—Part 1: Oak and beech. EN 975-1. Brussels, Belgium: CEN.

Google Scholar

CEN (European Committee for Standardization). 2012. Timber structures—Structural timber and glued laminated timber—Determination of some physical and mechanical properties. EN 408. Brussels, Belgium: CEN.

Google Scholar

CEN (European Committee for Standardization). 2014. Eurocode 5: Design of timber structures—Part 1-1: General—Common rules and rules for buildings. EN 1995-1-1. Brussels, Belgium: CEN.

Google Scholar

CEN (European Committee for Standardization). 2016. Timber structures—Calculation and verification of characteristic values. EN 14358. Brussels, Belgium: CEN.

Google Scholar

CEN (European Committee for Standardization). 2018. Structural timber—Determination of characteristic values of mechanical properties and density. EN 384. Brussels, Belgium: CEN.

Google Scholar

Chen, S. S., and R. A. Gopinath. 2000. “Gaussianization.” In Proc., Advances in Neural Information Processing Systems 13, Papers from Neural Information Processing Systems (NIPS) 2000, 423–429. Cambridge, MA: MIT Press.

Google Scholar

Deodatis, G., and R. C. Micaletti. 2001. “Simulation of highly skewed non-Gaussian stochastic processes.” J. Eng. Mech. 127 (12): 1284–1295. https://doi.org/10.1061/(ASCE)0733-9399(2001)127:12(1284).

Google Scholar

DIN (German Institute for Standardization). 2008. Strength grading of wood—Part 5: Sawn hard wood. DIN 4074-5. Berlin: DIN.

Google Scholar

Duxbury, P. M., P. L. Leath, and P. D. Beale. 1987. “Breakdown properties of quenched random systems: The random-fuse network.” Phys. Rev. B 36 (1): 367–380. https://doi.org/10.1103/PhysRevB.36.367.

Google Scholar

Ehlbeck, J., F. Colling, and R. Görlacher. 1984. Einfluß keilgezinkter Lamellen auf die Biegefestigkeit von Brettschichtholzträgern. [In German.]. Karlsruhe, Germany: Versuchsanstalt für Stahl, Holz und Steine, Universität Fridericiana Karlsruhe.

Google Scholar

Faye, C., G. Legrand, D. Reuling, and J. D. Lanvin. 2017. “Experimental investigations on the mechanical behaviour of glued laminated beams made of oak.” In Proc., Int. Network on Timber Engineering Research (INTER)—Proc. Meeting 50, 193–206. Karlsruhe, Germany: Timber Scientific Publishing.

Google Scholar

Fink, G. 2014. “Influence of varying material properties on the load-bearing capacity of glued laminated timber.” Ph.D. thesis, Institute of Structural Engineering, ETH Zurich.

Google Scholar

Foschi, R. O., and J. D. Barrett. 1980. “Glued-laminated beam strength: A model.” J. Struct. Div. 106 (8): 1735–1754. https://doi.org/10.1061/JSDEAG.0005496.

Google Scholar

Freas, A. D., and M. L. Selbo. 1954. Fabrication and design of glued laminated wood structural members. Madison, WI: Dept. of Agriculture, Forest Service, Forest Products Laboratory.

Google Scholar

Frese, M. 2006. “Die Biegefestigkeit von Brettschichtholz aus Buche. Experimentelle und numerische Untersuchungen zum Laminierungseffekt.” [In German.] Ph.D. thesis, Karlsruher Berichte zum Ingenieurholzbau, Universität Karlsruhe—Holzbau und Baukonstruktionen.

Google Scholar

Grigoriu, M. 1984. “Crossings of non-Gaussian translation processes.” J. Eng. Mech. 110 (4): 610–620. https://doi.org/10.1061/(ASCE)0733-9399(1984)110:4(610).

Google Scholar

Isaksson, T. 1999. “Modelling the variability of bending strength in structural timber—Length and load configuration effects.” Ph.D. thesis, Div. of Structural Engineering, Lund Institute of Technology.

Google Scholar

Kaplan, E. L., and P. Meier. 1958. “Nonparametric estimation from incomplete observations.” J. Am. Stat. Assoc. 53 (282): 457–481. https://doi.org/10.1080/01621459.1958.10501452.

Google Scholar

Kline, D., F. Woeste, and B. Bendtsen. 1986. “Stochastic model for modulus of elasticity of lumber.” Wood Fiber Sci. 18 (2): 228–238.

Google Scholar

Lam, F., and E. Varoglu. 1990. “Effect of length on the tensile strength of lumber.” For. Prod. J. 40 (5): 37–42.

Google Scholar

Lam, F., and E. Varoglu. 1991a. “Variation of tensile strength along the length of lumber—Part 1: Experimental.” Wood Sci. Technol. 25 (5): 351–359. https://doi.org/10.1007%2FBF00226174.

Google Scholar

Lam, F., and E. Varoglu. 1991b. “Variation of tensile strength along the length of lumber—Part 2: Model development and verification.” Wood Sci. Technol. 25 (6): 449–458. https://doi.org/10.1007%2FBF00225237.

Google Scholar

Le, J.-L., and Z. P. Bažant. 2011. “Unified nano-mechanics based probabilistic theory of quasibrittle and brittle structures: II. Fatigue crack growth, lifetime and scaling.” J. Mech. Phys. Solids 59 (7): 1322–1337. https://doi.org/10.1016/j.jmps.2011.03.007.

Google Scholar

Le, J.-L., Z. P. Bažant, and M. Z. Bažant. 2011. “Unified nano-mechanics based probabilistic theory of quasibrittle and brittle structures: I. Strength, static crack growth, lifetime and scaling.” J. Mech. Phys. Solids 59 (7): 1291–1321. https://doi.org/10.1016/j.jmps.2011.03.002.

Google Scholar

Le, J.-L., A. C. Falchetto, and M. O. Marasteanu. 2013. “Determination of strength distribution of quasibrittle structures from mean size effect analysis.” Mech. Mater. 66 (Nov): 79–87. https://doi.org/10.1016/j.mechmat.2013.07.003.

Google Scholar

Madsen, B. 1990. “Length effects in 38 mm spruce-pine-fir dimension lumber.” Can. J. Civ. Eng. 17 (2): 226–237. https://doi.org/10.1139/l90-028.

Google Scholar

Mogensen, P. K., and A. N. Riseth. 2018. “Optim: A mathematical optimization package for Julia.” J. Open Source Software 3 (24): 615. https://doi.org/10.21105/joss.00615.

Google Scholar

Nelson, W., and G. J. Hahn. 1972. “Linear estimation of a regression relationship from censored data: Part I—Simple methods and their application.” Technometrics 14 (2): 247–269. https://doi.org/10.1080/00401706.1972.10488912.

Google Scholar

Newlin, J. A., and G. W. Trayer. 1924. The influence of the form of a wooden beam on its stiffness and strength, II: Form factors of beams subjected to transverse loading only. Washington, DC: National Advisory Committee for Aeronautics.

Google Scholar

Odell, P. M., K. M. Anderson, and R. B. D’Agostino. 1992. “Maximum likelihood estimation for interval-censored data using a Weibull-based accelerated failure time model.” Biometrics 48 (3): 951–959. https://doi.org/10.2307/2532360.

Google Scholar

Olsson, A., and J. Oscarsson. 2017. “Strength grading on the basis of high resolution laser scanning and dynamic excitation: A full scale investigation of performance.” Eur. J. Wood Wood Prod. 75 (1): 17–31. https://doi.org/10.1007/s00107-016-1102-6.

Google Scholar

Pang, S.-D., Z. P. Bažant, and J.-L. Le. 2008. “Statistics of strength of ceramics: Finite weakest-link model and necessity of zero threshold.” Int. J. Fract. 154 (1–2): 131–145. https://doi.org/10.1007/s10704-009-9317-8.

Google Scholar

Rinne, H. 2009. The Weibull distribution—A handbook. Boca Raton, FL: CRC Press.

Google Scholar

Rouger, F., and J. D. Barrett. 1995. “Size effects in timber.” In STEP 3, edited by F. Holz, 1–24. Düsseldorf, Germany: Arbeitsgemeinschaft.

Google Scholar

Showalter, K. L., F. E. Woeste, and B. A. Bendtsen. 1987. Effect of length on tensile strength in structural lumber. Madison, WI: USDA, Forest Service, Forest Products Laboratory.

Google Scholar

Tapia, C., and S. Aicher. 2018. “Modelling the variation of mechanical properties along oak boards.” In Proc., Int. Network on Timber Engineering Research (INTER)—Meeting 51, 31–45. Karlsruhe, Germany: Timber Scientific Publishing.

Google Scholar

Tapia, C., and S. Aicher. 2019. “Variation and serial correlation of modulus of elasticity between and within European oak boards (Quercus robur and Q. petraea).” Holzforschung 74 (1): 33–46. https://doi.org/10.1515/hf-2019-0038.

Google Scholar

Tapia, C., and S. Aicher. 2021a. Replication data for: Survival analysis of tensile strength variation and simulated length-size effect along oak boards. Stuttgart, Germany: Univ. of Stuttgart.

Google Scholar

Tapia, C., and S. Aicher. 2021b. “Simulation of the localized modulus of elasticity of hardwood boards by means of an autoregressive model.” J. Mater. Civ. Eng. 33 (6): 04021132. https://doi.org/10.1061/(ASCE)MT.1943-5533.0003696.

Google Scholar

Taylor, S., and D. Bender. 1988. “Simulating correlated lumber properties using a modified multivariate normal approach.” Trans. ASAE 31 (1): 182–186. https://doi.org/10.13031/2013.30685.

Google Scholar

Taylor, S., and D. Bender. 1991. “Stochastic model for localized tensile strength and modulus of elasticity in lumber.” Wood Fiber Sci. 23 (4): 501–519.

Google Scholar

van den Born, I. C., A. Santen, H. D. Hoekstra, and J. T. M. D. Hosson. 1991. “Mechanical strength of highly porous ceramics.” Phys. Rev. B 43 (4): 3794–3796. https://doi.org/10.1103/PhysRevB.43.3794.

Google Scholar

Weibull, W. 1951. “A statistical distribution function of wide applicability.” J. Appl. Mech. 18 (3): 293–297. https://doi.org/10.1115/1.4010337.

Google Scholar

Zhou, H., H. Ren, J. Lu, J. Jiang, and X. Wang. 2010. “Length effect on the tension strength between mechanically graded high-and low-grade Chinese fir lumber.” For. Prod. J. 60 (2): 144–149. https://doi.org/10.13073/0015-7473-60.2.144.

Google Scholar

Information & Authors

Information

Published In

Journal of Engineering Mechanics

Volume 148 • Issue 1 • January 2022

Copyright

This work is made available under the terms of the Creative Commons Attribution 4.0 International license, https://creativecommons.org/licenses/by/4.0/.

History

Received: Feb 18, 2021

Accepted: Jul 15, 2021

Published online: Nov 2, 2021

Published in print: Jan 1, 2022

Discussion open until: Apr 2, 2022

Authors

Affiliations

Cristóbal Tapia https://orcid.org/0000-0003-2228-1686 [email protected]

Materials Testing Institute, Univ. of Stuttgart, Pfaffenwaldring 4b, Stuttgart 70569, Germany (corresponding author). ORCID: https://orcid.org/0000-0003-2228-1686. Email: [email protected]

View all articles by this author

Simon Aicher https://orcid.org/0000-0003-1759-9884

Chief Academic Director and Head of Department, Materials Testing Institute, Univ. of Stuttgart, Pfaffenwaldring 4b, Stuttgart 70569, Germany. ORCID: https://orcid.org/0000-0003-1759-9884

View all articles by this author

Metrics & Citations

Metrics

Citations

Download citation

If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.

Cited by

Stephen M. Walley, Samuel J. Rogers, Is Wood a Material? Taking the Size Effect Seriously, Materials, 10.3390/ma15155403, 15, 15, (5403), (2022).
Crossref

Abstract

Introduction

Materials and Methods

Oak Boards, Grading, Global and Local MOE

Multiple Strength Measurements

Finite Weakest Link Theory

Application of Survival Analysis to the Tensile Strength Results

Model for the Simulation of Tensile Strength Profiles

Order Statistics and Extreme Value Theory

Results and Discussion

Observed Variation of Tensile Strength for All Boards

Estimated Parameters of the Local Tensile Strength Models

Verification of the Parametric Models by Extreme Value Theory

Simulation of Tensile Strength Profiles along Board Length

Simulation of Length Effect for Tensile Strength

Discussion

Conclusions

Notation

Data Availability Statement

Acknowledgments

References

Information

Published In

Copyright

History

Authors

Affiliations

Metrics

Citations

Download citation

Cited by

Figures

Other

Share

Copy the content Link

Share with email

Share

Request Username

Create a new account

Change Password

Password Changed Successfully

Verify Phone

Congrats!